The earliest, and still most influential, geometric structure which emerged as a candidate for vacuum configuration of unified string theories was that of a Calabi-Yau manifold, equipped with a holomorphic vector bundle admitting a Hermitian-Einstein metric. This was proposed by Candelas, Horowitz, Strominger, and Witten [1]} for the heterotic string, and it could rely on the fundamental existence theorems for such structures proved earlier by Yau [2]} and Donaldson-Uhlenbeck-Yau [3]}, [4]}. But other geometric structures have emerged since, motivated by the other string theories, for which analogous existence theorems are not yet available. Such theorems can be expected to require the full theory of non-linear partial differential equations and, as advocated in [5]}, [6]}, notably the theory of geometric flows. In particular, flow approaches have been proposed in [5]} for the Hull-Strominger system in heterotic string theory, in [6]}, [9]} for the Type IIB string, in [10]} for the Type IIA string, and in [11]} for 11-dimensional supergravity.
